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Hamiltonians in the quantum field theory can be regarded as self-adjoint operators on Hilbert spaces. I analyze the spectrum of the Hamiltonian non-pertubatively. Perturbations of embedded eigenvalues and ultraviolet and infrared divergences are the subtle problems. The existence and absence of ground states, the multiplicity of ground states, spectral scattering theory, resonances and renormalizations are studied by using operator theory, functional integrals, Gibbs measures, the theory of one-parameter semi-groups and renormalization group.